与微分算子相联系的奇异积分算子的最佳加权模估计及其应用

The relationship between the weighted norm of operator and the weighted function played an important role in harmonic analysis and in partial differential equations. Recently, the sharp Ap weighted estimates for classical Calderon–Zygmund singular integral operators have been attracting attentions of many research workers. In S. Petermichl, T. Hytonen, C. Perez, S. Treil and A. Volberg''s efforts, the sharp Ap weighted estimates for classical Calderon–Zygmund singular integral operator have been absolutely solved. And A. Lerner obtained the sharp Ap weighted estimates for classical Littlewood-Paley function. This Program will consider the sharp Ap weighted estimates for Littlewood-Paley function, Riesz transform and singular integral operators associated with differential operators and applications.

算子的加权模和权函数之间的依赖关系在调和分析领域和微分方程领域都有着重要的应用。近年来有关经典Calderon–Zygmund奇异积分算子的最佳Ap权估计问题引起了学者们的广泛注意。在S. Petermichl、T. Hytonen、C. Perez、S. Treil和A. Volberg等人的推动下，经典Calderon–Zygmund奇异积分算子的最佳Ap权估计问题彻底得到解决。A. Lerner在研究该问题的时候还得到了经典Littlewood-Paley函数的最佳Ap权估计。本项目将在此基础上考虑与微分算子相联系的Littlewood-Paley函数、Riesz变换和一般的奇异积分算子的最佳Ap权估计及其应用。

算子的加权模和权函数之间的依赖关系在调和分析领域和微分方程领域都有着重要的应用。近年来，经典Calderon–Zygmund 奇异积分算子及经典Littlewood-Paley 函数的最佳Ap权估计问题彻底得到解决。本项目在此基础上研究了与微分算子相联系的Littlewood-Paley 函数及、Hormander 型谱乘子的最佳加权估计问题。本项目得到了一类与微分算子相联系的Littlewood-Paley 函数的最佳Ap权估计，并且利用与微分算子相联系的Littlewood-Paley 理论证明了Hormander型谱乘子在加权LP空间上的有界性。